Lecture 1: Examples, connectedness, paths and cycles

Transcription

1 Lecture 1: Examples, connectedness, paths and cycles Anders Johansson lör

2 Outline The course plan Examples and applications of graphs Relations The definition of graphs as relations Connectedness, paths and cycles

3 Topic The course plan Examples and applications of graphs Relations The definition of graphs as relations Connectedness, paths and cycles

4 Literature, teaching format and examination Literature Grimaldi, Ralph P.; Discrete and combinatorial mathematics, 5th edition, Pearson Addison Wesley, Chapters Extra material handed out. (Not too demanding) Teaching format: We have twenty combined lectures and tutorials. Weight will be put on problem solving. Examination We will have four problem sessions, where you can present your own solutions to problems. Problem sheets will be handed out and posted at least a week in advance. Details about dates etc., will be posted on the course page later. Participation in the problem sessions (at least 2/4) will give bonus at the written exam in December; the first exam-question will automatically be graded with full points.

5 Lesson and exercise planning No Section Content Proposed exercises Definitions and examples, paths and cycles 2,4,6,7,8,10,13,14,15, Subgraphs and graph-homomorphisms 1,2,4,5,7,8,10,11, Degrees, parity and euler tours 1,2,3,5,8,11,13,16,26,27,30,32,33, (+) Planar graphs, cycles and cuts 1,2,11,13,17,19,21,22,23,27, Hamiltonian graphs 1,2,4,10,12,18,20,21,25, Chromatic properties 1,2,4,7,8,9,14,16,17,18, (+) More about colourings S11 17,18,21,22, (+) Acyclic graphs and trees 2,4,5,7,9,10,11,18,19,21,22,23, Rooted trees and search 1,2,5,7,9,10,12,14,15,16,19, Trees and sorting 1,3, Trees and codes 1,3,6, Decomposition into blocks 1,4,5,6,7,8,12,13 S12 5,6,7,8,11, Shortest path problems 1,2,4,5,6,7, Greedy algorithms and minimum spanning 1,2,4,5,6,7,9 tree Transport networks 1,3,6 (+) (+) Transport problems Matchings in bipartite graphs 1,2,4,5,7,13 S13 5,6,8, (+) Duality principles in linear programming 19 (+) Random graphs and random graph statistics 20 Tutorial (buffer)

6 Topic The course plan Examples and applications of graphs Relations The definition of graphs as relations Connectedness, paths and cycles

7 A simple symmetric relation (undirected graphs) (a) A symmetric graph on 30 vertices (b) Another 3-regular graph Figure: Undirected simple graphs: Main object

8 Automata and state transition graphs Directed graphs, where arrows stand for a transition in time between states are also very common: Markov chains, automata, game theory, C 0 B 0 start 1 A F E D G 1

9 Dependency graphs, factor graphs, etc. Many application of graphs starts from graphs where nodes representing variables are connected if they are dependent in some way. Examples are Bayesian networks, graphical models, factor graphs,... Figure: A Baeysian network

10 Relationship diagrams, data base models,... In Computer Science the use of graphs has a long tradition. Here is an example of a entity relation diagram used in database constructions. name name n C-I 1 institute 1 code course n S-C m S-I n student name number grade

11 Topic The course plan Examples and applications of graphs Relations The definition of graphs as relations Connectedness, paths and cycles

12 Naive set theory Examples of sets are {1, 2}, {1, 1, 2} = {1, 2}, {3n Z : n is prime }, {{1, 3}, {2}}, etc. The statement that x is an element of A, is written x A. Two sets A and B are equal when they have precisely the same elements, that is, x A x B. A set encode as a geometric object a unary relation on objects or a proposition about outcomes. Logic is obtained by set operations. x A B A B A \ B

13 Sets of sets and the power set 2 X Given a set X, we obtain the power set 2 X consisting of all subsets of X. The set ( X k ) is the set of all k-subsets of X. If X = {1, 2, 3} then 2 X = {, {1}, {2}, {3}, {1, 2}, {2, 3}, {3, 1}, {1, 2, 3}}. and ( ) X = {{1, 2}, {2, 3}, {3, 1}} and 2 ( ) X = {{1, 2, 3}}. 3 Note the cardinalities (=number of elements) ( ) ( ) 2 X = 2 X, X X =. k k

14 The Cartesian product of sets Given two sets A and B, the Cartesian product, A B is the set of pairs (two-tuples) (a, b) such that a A and b B. For example, given the set of playing card ranks and suits and R = {A, K, Q, J, 10, 9,..., 3, 2} S = {,,, } we obtain the 52-element set R S = {(A, ), (K, ),..., (2, ), (A, ),..., (3, ), (2, )}. A poker-hand is thus a 5-subset of R S, i.e. an element in ( ) R S J Q K A A K Q J

15 The Cartesian product of sets Given two sets A and B, the Cartesian product, A B is the set of pairs (two-tuples) (a, b) such that a A and b B. For example, given the set of playing card ranks and suits and R = {A, K, Q, J, 10, 9,..., 3, 2} S = {,,, } we obtain the 52-element set R S = {(A, ), (K, ),..., (2, ), (A, ),..., (3, ), (2, )}. A poker-hand is thus a 5-subset of R S, i.e. an element in ( ) R S J Q K A A K Q J Full house

16 Relations: Subsets of a cartesian product A B A (binary) relation R between elements of A and elements of B is (can be interpreted as) a subset R A B, where arb precisely if (a, b) R. d c b a If A = B = X, we say a relation Q X X is a relation on X. c b d X a e d

17 Functions A relation f A B is a function from A to B if for every a A there is exactly one element in B related to a. This element is written f (a). d c A f B A function is surjective if every element in B is related to at least one element in A. b a A function is injective if no element in B is related to more than one element in A. d A g f (a) B 6 A bijective function f : A B is one which is both surjective and injective. In this case we have B = f (A) = A. c b a

18 Symmetric, transitive and reflexive relations A relation Q on X is symmetric if, for all x, y X, xqy implies yqx. It is reflexive if xqx for all x X. d c b a A relation Q on X is transitive if, for all x, y, z X, e f xqy yqz = xqz. A symmetric, reflexive and transitive relation is an equivalence relation.

19 Symmetric, transitive and reflexive relations A relation Q on X is symmetric if, for all x, y X, xqy implies yqx. It is reflexive if xqx for all x X. d c b a A relation Q on X is transitive if, for all x, y, z X, e f xqy yqz = xqz. A symmetric, reflexive and transitive relation is an equivalence relation.

20 Symmetric, transitive and reflexive relations A relation Q on X is symmetric if, for all x, y X, xqy implies yqx. It is reflexive if xqx for all x X. d c b a A relation Q on X is transitive if, for all x, y, z X, e f xqy yqz = xqz. c b A symmetric, reflexive and transitive relation is an equivalence relation. d a e f

21 Symmetric, transitive and reflexive relations A relation Q on X is symmetric if, for all x, y X, xqy implies yqx. It is reflexive if xqx for all x X. d c b a A relation Q on X is transitive if, for all x, y, z X, e f xqy yqz = xqz. c b A symmetric, reflexive and transitive relation is an equivalence relation. d a e f

22 Topic The course plan Examples and applications of graphs Relations The definition of graphs as relations Connectedness, paths and cycles

23 Definition of simple directed graphs Def: Let V be a set and E V V a binary relation on V. A simple digraph G is the pair G = (V, E), where V is called the set of vertices and E the set of directed edges or arcs. We say i V is adjacent to j V if ij = (i, j) E and adjacent from k V if (k, i) E. The edge (i, j) E is an out-edge at i and an in-edge at j; j is an out-neighbour of i and i is an in-neighbour of j If E does not include edges of the form (i, i), the graph is without loops.

24 Definition of simple directed graphs Def: Let V be a set and E V V a binary relation on V. A simple digraph G is the pair G = (V, E), where V is called the set of vertices and E the set of directed edges or arcs. We say i V is adjacent to j V if ij = (i, j) E and adjacent from k V if (k, i) E. The edge (i, j) E is an out-edge at i and an in-edge at j; j is an out-neighbour of i and i is an in-neighbour of j If E does not include edges of the form (i, i), the graph is without loops.

25 Undirected simple graph Def: An undirected simple graph is a pair G = (V, E), where direction E ( V 2). We often write elements {i, j} of E as ij direction is disregarded. An undirected graph corresponds to a symmetric adjacency relation; i adjacent to j and vice versa precisely if {i, j} E. This correspond to an underlying symmetric reflexive or anti-reflexive relation We say that a vertex v is incident with an edge e E if v e.

26 General multigraphs; head and tail functions; incidence relation Let V and E be two finite sets. A general (multi-) digraph G = (V, E, head tail) is given by two functions head : E V and tail : E V. The multiplicity of an edge (u, v) V V is the number edges e such that head e = u and tail e = v. The incidence relation, Inc, from E to V, is defined as follows: e is incident with v if v = head e or v = tail e. An undirected multi-graph G = (V, E) is an equivalence class with a fixed incidence relation. Multiplicity of edge {u, v} is the number of edges such that Inc({e}) = {u, v}. Every digraph G with the same incidence relation is called an orientation of G. E head tail e 1 A B e 2 A B e 3 B C e 4 C D e 5 B E e 6 E F e 7 B E e 8 A F e 9 E D F E e 8 e e e e A e 7 1 D e 2 e 4 e 3 B C

27 General multigraphs; head and tail functions; incidence relation Let V and E be two finite sets. A general (multi-) digraph G = (V, E, head tail) is given by two functions head : E V and tail : E V. The multiplicity of an edge (u, v) V V is the number edges e such that head e = u and tail e = v. The incidence relation, Inc, from E to V, is defined as follows: e is incident with v if v = head e or v = tail e. An undirected multi-graph G = (V, E) is an equivalence class with a fixed incidence relation. Multiplicity of edge {u, v} is the number of edges such that Inc({e}) = {u, v}. Every digraph G with the same incidence relation is called an orientation of G. E head tail e 1 A B e 2 A B e 3 B C e 4 C D e 5 B E e 6 E F e 7 B E e 8 A F e 9 E D F E e 8 e e e e A e 7 1 D e 2 e 4 e 3 B C

28 General multigraphs; head and tail functions; incidence relation Let V and E be two finite sets. A general (multi-) digraph G = (V, E, head tail) is given by two functions head : E V and tail : E V. The multiplicity of an edge (u, v) V V is the number edges e such that head e = u and tail e = v. The incidence relation, Inc, from E to V, is defined as follows: e is incident with v if v = head e or v = tail e. An undirected multi-graph G = (V, E) is an equivalence class with a fixed incidence relation. Multiplicity of edge {u, v} is the number of edges such that Inc({e}) = {u, v}. Every digraph G with the same incidence relation is called an orientation of G. E head tail e 1 A B e 2 A B e 3 B C e 4 C D e 5 B E e 6 E F e 7 B E e 8 A F e 9 E D F E e 8 e e e e A e 7 1 D e 2 e 4 e 3 B C

29 The Cashier/Guard problem Consider the following (simple, undirected) graph. (The normal case!) a b c d e f g h i j k Figure: A graph of cashiers connected by aisles Place a minimum set of guards so that each cashier has at least one guard in some neighbouring aisle. Find a minimum set S such that every vertex is either in S or adjacent to some vertex in S. What is the minimum size of a set S V, such that every edge is incident with at least one vertex in S?

30 Topic The course plan Examples and applications of graphs Relations The definition of graphs as relations Connectedness, paths and cycles

31 Walks and reachability The problem of Reachability Can we reach vertex v from vertex u by walking around in the graph? If so, in how many ways can we do this? Def: An uv-walk W of length n in the (multi-di-) graph G = (V, E) is an alternating sequence of vertices x i V, i = 0, 1,..., n, and edges e j E, j = 1,..., n, W : u = x 0, e 1, x 1, e 2, x 2,..., e n 1, x n 1, e n, x n = v, such that, for all i = 1,..., n, both x i 1 and x i are incident with e i. If for all i, x i 1 = head e i and x i = tail e i it is a directed walk otherwise it is an oriented walk. If G is simple then a walk W is specified by the sequence W := (x 0, x 1,..., x n ) V := V V 2 V n....

32 Closed walks, trails, circuits, paths and cycles An uv-walk is u = x 0, x 1,..., x n = v, closed if a = b a trail if all edges are distinct a circuit if it is a closed trail a path a cycle if all vertices are distinct if it is a closed path c 4 b 5 b 4 a 5 c 3 a 4 c 5 a 6 b6 c 6 a 7 b 7 c 7 a 3 b 3 a 8 c 2 a 2 b 2 a 1 b 1 a 9 b 8 c 8 c 1 a 0 b 0 c 0 b 9 c 9

33 Closed walks, trails, circuits, paths and cycles An uv-walk is u = x 0, x 1,..., x n = v, closed if a = b a trail if all edges are distinct a circuit if it is a closed trail a path a cycle if all vertices are distinct if it is a closed path c 4 b 5 b 4 a 5 c 3 a 4 c 5 a 6 b6 c 6 a 7 b 7 c 7 a 3 b 3 a 8 c 2 a 2 b 2 a 1 b 1 a 9 b 8 c 8 c 1 a 0 b 0 c 0 b 9 c 9 A a 0 a 2 -walk

34 Closed walks, trails, circuits, paths and cycles An uv-walk is u = x 0, x 1,..., x n = v, closed if a = b a trail if all edges are distinct a circuit if it is a closed trail a path a cycle if all vertices are distinct if it is a closed path c 4 b 5 b 4 a 5 c 3 a 4 c 5 a 6 b6 c 6 a 7 b 7 c 7 a 3 b 3 a 8 c 2 a 2 b 2 a 1 b 1 a 9 b 8 c 8 c 1 a 0 b 0 c 0 b 9 c 9 A a 0 a 2 -trail

35 Closed walks, trails, circuits, paths and cycles An uv-walk is u = x 0, x 1,..., x n = v, closed if a = b a trail if all edges are distinct a circuit if it is a closed trail a path a cycle if all vertices are distinct if it is a closed path A a 0 a 2 -path c 4 b 5 b 4 a 5 c 3 a 4 c 5 a 6 b6 c 6 a 7 b 7 c 7 a 3 b 3 a 8 c 2 a 2 b 2 a 1 b 1 a 9 b 8 c 8 c 1 a 0 b 0 c 0 b 9 c 9

36 Closed walks, trails, circuits, paths and cycles An uv-walk is u = x 0, x 1,..., x n = v, closed if a = b a trail if all edges are distinct a circuit if it is a closed trail a path a cycle if all vertices are distinct if it is a closed path A a 0 a 0 -circuit c 4 b 5 b 4 a 5 c 3 a 4 c 5 a 6 b6 c 6 a 7 b 7 c 7 a 3 b 3 a 8 c 2 a 2 b 2 a 1 b 1 a 9 b 8 c 8 c 1 a 0 b 0 c 0 b 9 c 9

37 Closed walks, trails, circuits, paths and cycles An uv-walk is u = x 0, x 1,..., x n = v, closed if a = b a trail if all edges are distinct a circuit if it is a closed trail a path a cycle if all vertices are distinct if it is a closed path A a 0 a 0 -cycle c 4 b 5 b 4 a 5 c 3 a 4 c 5 a 6 b6 c 6 a 7 b 7 c 7 a 3 b 3 a 8 c 2 a 2 b 2 a 1 b 1 a 9 b 8 c 8 c 1 a 0 b 0 c 0 b 9 c 9

38 Compute the number of closed walks in a lollipop graph Consider the following graph with a loop. v w Figure: The lollipop graph Determine a recursion formula for a n the number of closed vv-walks in this graph.

39 Paths determine the connectivity structure We can describe a path/cycle in any type of graph as a subset of edges and (possibly) the starting point. Paths and cycles are also subgraphs. (More next time) Theorem (Paths are enough) There exists an (directed/oriented) uv-walk if and only if there exists a (directed/oriented) uv-path. Task: Prove this!

40 Components, connectedness A vertex u V is reachable from v V if there exists an oriented/directed uv-walk or a path. For oriented walks/paths for undirected graphs this is an equivalence relation on V. The equivalence classes are called connected components. The number of components of a graph G is denoted by c(g). (κ(g) in the book) A graph is connected if c(g) = 1. For digraphs we can say that two vertices u an v are strongly connected if there are both a directed uv-walk and a directed vu-walk. This is again an equivalence relation; partitioning V into strong components c(g) = 3. 1

41 Components, connectedness A vertex u V is reachable from v V if there exists an oriented/directed uv-walk or a path. For oriented walks/paths for undirected graphs this is an equivalence relation on V. The equivalence classes are called connected components. The number of components of a graph G is denoted by c(g). (κ(g) in the book) A graph is connected if c(g) = 1. For digraphs we can say that two vertices u an v are strongly connected if there are both a directed uv-walk and a directed vu-walk. This is again an equivalence relation; partitioning V into strong components Strong components 1

42 Distance, diameter and girth Given a graph G = (V, E) the length (or + ) of a shortest path between u and v, u, v V, is called the distance between u and v. It is denoted dist G (u, v). The diameter, diam(g), of a graph is the maximum distance between two vertices. 1. What is the diameter and girth of the Petersen graph? 2. Prove that for every non-acyclic graph g(g) 2 diam(g) + 1. Figure: The Petersen graph The girth, g(g), of a graph is the length of the smallest cycle. (If the graph contains no cycles it is called acyclic.)